Problem: Find the solutions to
\[\frac{13x - x^2}{x + 1} \left( x + \frac{13 - x}{x + 1} \right) = 42.\]Enter all the solutions, separated by commas.
Answer: Multiplying both sides by $(x + 1)^2,$ we get
\[(13x - x^2)(x(x + 1) + (13 - x)) = 42(x + 1)^2.\]This expands to $x^4 - 13x^3 + 55x^2 - 85x + 42 = 0,$ which factors as $(x - 1)(x - 6)(x^2 - 6x + 7) = 0.$  By the quadratic formula, the roots of $x^2 - 6x + 7 = 0$ are $3 \pm \sqrt{2}.$  Therefore, the solutions are $\boxed{1, 6, 3 + \sqrt{2}, 3 - \sqrt{2}}.$